Two Hop Sparse Grooming Network Design

Nh/W 3 2 1 0 389 365 292 1 413 402 306 Ng Nλ LP + FG 6 3 LP + SG 1 3 LP + NG 0 4 Nh Nλ LT + NH 0 3 LT + SH 1 3 LT + FH 6 3 (a) (b) (c)

Figure 4.5 Results from ILP for the six-node network (a) Maximize throughput formulation with δ = 2 (b) Minimize cost formula- tion for lightpaths. LP = lightpaths, SG = sparse grooming, FG = full grooming, NG = no grooming (c) Minimize cost for- mulation for light-trails. LT + NH = light-trails with no trail length constraints and no hubs, LT + SH = light-trails with sparse hub nodes, δ = 2, LT + FH = light-trails with all nodes hub capable, δ = 2

4.2.1 Simulation Results

In this section, we present the results for ILP that was formulated in Section 3.4.2 and for heuristics PC, RC and EC that were presented in 3.4.3. The networks are assumed to use light-trail circuits to carry traffic. The capacity of a wavelength is arbitrarily assigned to be 48 units.

4.2.1.1 ILP Results

The ILP formulation in the max throughput form and the min cost form were solved using CPLEX 8.1.0 for the network shown in Figure 4.1(a). The requests between any node pair are of three granularities - 1, 3 and 12 units. With probability p, a connection of each granularity is established between a node pair and we set p = 0.5 for the illustrative example in Figure 4.5. The number of such 1, 3 and 12 streams are uniformly distributed between (0,1), (0,1) and (2,3) units respectively.

For the throughput maximization ILP problem in light-trails with δ = 2, if Nh = 0 is added

as an additional constraint, only 389 units are carried for W =3 or more as shown in Figure 4.5(a). If we set Nh=1, all the traffic (413 units) can be supported for W = 3 or more. At

present in the network showing that wavelength is the bottleneck.

For the cost minimization ILP problem, we assume Cλ=1 and Ch = 1, and provide all pos-

sible paths as input. We find that lightpaths with no grooming requires 4 wavelengths whereas lightpaths with full grooming require 3 wavelengths as shown in Figure 4.5(b). Lightpaths with sparse grooming requires only one grooming node and needs three wavelengths. This shows that sparse grooming can achieve performance close to full grooming. When light-trails with no trail length constraints are studied, no hubbing was required and still only 3 wavelengths were consumed. When the trail length limit constraint is imposed and trails of length 2 are provided as input, all the traffic is still carried using 3 wavelengths while one node is designated as a hub node as shown in Figure 4.5(c).

4.2.1.2 Heuristic Results

We apply our heuristics to study the effect of sparse hubbing on a 25-node light-trail network shown in Figure 4.1(b). The number of streams of sizes 1, 3 and 12 between a node pair are uniformly distributed as (0,12), (0,2) and (0,1) respectively. We set W = 13, δ = 3, and observe average throughput of the network as a function of the number of H-nodes in Figure 4.6(a) by running the simulation with 500 different traffic matrices having the above distribution. When Nh = 0, about 74 % of the traffic is carried while the rest are physically

blocked. With only a few H-nodes, the throughput climbs steeply and reaches close to 100 %. We observe that the PC heuristic yields the best throughput closely followed by the EC heuristic. The RC heuristic performs the worst but they all converge to the same value as the number of H-nodes increase.

We study the average wavelength requirement by running the simulation for 2000 traffic matrices of above distribution in Figure 4.6(b). In this case, if we do not observe 100 % throughput, it is because of physically blocked node pairs and not because of wavelength exhaustion. We observe that EC heuristic yields the minimum number of wavelengths closely followed by the PC heuristic. The random heuristic, on an average, is unable to achieve 100 % throughput until about Nh = 11 (not shown in figure), while the other two yield 100 %

(a) (b)

Figure 4.6 (a) Network throughput as a function of the number of hub nodes for the 25 node network with W = 13, δ = 3 (b) Wave- length requirements for 100 % throughput (throughput is less for RC heuristic) at δ = 3, as a function of the number of hub nodes for the 25 node network

throughput with only just one hub node. Since the hubs are randomly chosen in RC heuristic, the chosen hubs may not be in the proximity of every physically blocked node pair. In both the PC and EC heuristic, the first node that is chosen corresponds to the center of the graph. Since the diameter is 6, and the center of the graph has the longest path only of 3 from it, the first chosen vertex is in the proximity of every physically blocked pair. If wavelength availability is not a bottleneck, it can serve as the hub for all physically blocked pairs. Hence, 100 % throughput can be observed even with one hub node.

As wavelength requirements decrease with increase in number of hub nodes, it may be interesting to identify the exact number of hub nodes required for a given traffic scenario. We can identify the network cost similar to the approach in [147].

Define the ratio ρ,

ρ = Ch Cλ

The cost of the network Cn is

Cn= Nh× Ch+ Nλ× Cλ = (Nh× ρ + Nλ) × Cλ

(a) (b)

Figure 4.7 (a) Network costs plot to determine the optimal number of hub nodes (b) Wavelength utilization as a function of load for vary- ing trail length limits (D)

Cn= Nh× ρ + Nλ (4.1)

Figure 4.7(a) shows the cost of the network for various values of ρ. If the cost of the hub node is much larger than maintaining a wavelength, it may be better to operate with minimal number of hub nodes. For this specific example, if ρ = 1 or 2, the optimal cost is achieved at Nh=5 while for ρ = 0.1 or 0.2, the optimal cost is at Nh=7. The utilization of a wavelength

for various values of trail sizes are shown in Figure 4.7(b) . The load on x-axis corresponds to the parameter p described above. As p increases from 0 to 1, it can be seen that the wavelength utilization steadily increases and reaches about 82 % indicative of good packing by our heuristic. Heavier load at shorter δ is due to traffic rearrangement.

4.2.2 Conclusions

We studied the sparse hubbing problem in light-trail networks. We adopted a unified ap- proach for ILP formulation that is applicable for both groomed-lightpath and hubbed-light-trail networks, and presented results for a test network. We designed simple heuristics for H-node placement, traffic rearrangement and light-trail routing in the context of multiple granularity connections subject to non-bifurcation constraints. Our simulation results suggest that with only a small number of hub nodes, high network throughput and good wavelength utilization

can be achieved. Our research also gives guidelines for deciding the network operation point based on network element costs.